The OLS formula

Summary

• Given data $x_1,x_2,…,x_n$ and $y_1,y_2,…,y_n$ , we can add a linear trendline to the scatterplot of the points $\left( x_1,y_1 \right),…,(x_n,y_n)$ .
• The equation of a linear trendline is given by

$$y=b_1+b_2x$$

• We say that we use the OLS (Ordinary Least Squares) formula if we calculate the intercept $b_1$ and the slope $b_2$ using the following formulas:

$$b_2= \frac{\sum_{i=1}^{n}{ \left( x_i-\bar{x} \right)\left( y_i-\bar{y} \right) }}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }}= \frac{s_{x,y}^2}{s_x^2}$$

$$b_1=\bar{y}-b_2\bar{x}$$

• The linear trendline calculated using the OLS formula is called the OLS trendline .
• OLS in Excel: Linest(ydata,xdata,constant,statistics)